Team bats out of order:
Baker gets a hit.
Abel gets a hit, both Abel and Baker on base. (Baker at bat is now legal)
Charlie takes strike one. (Abel at bat is now legal)
Defensive team notices that Baker should have followed Abel and notifies the ump. Ump looks and agrees. Baker should be batting. But Baker is on base, so… the next legal batter would be Charlie. Who is batting illegally. But since the ump was notified, he is now legal.
Is this really how this could go down?
If so, imagine you are the offensive team. You notice your first two batters were out of order. The defensive team, for whatever reason, hasn’t appealed. Baker should be hitting after Abel, but doing that would clearly signal to the defensive team you screwed up since you’d need to tell the ump. You send Charlie to the plate and hope for the best. If Charlie gets a hit, you are probably screwed. If he bounces into a double play, the defensive team isn’t likely to take that off the board, so you are equally screwed. Both teams have an interest in letting Charlie hit. The first pitch goes by and now what? The offensive teams wants to tell the umpire about the screwup. Abel is now legal and can’t be called out. So the best strategy for the offensive team is to appeal and the defensive team wants to let Charlie keep hitting.
Baseball rules are really weird!
I once heard a sportscaster intone that baseball is the only sport that condones cheating. A runner leaving base early isn’t out until the defensive team appeals, even if the ump sees it. A batter out of order isn’t out until appealed to the ump, etc.
A runner leaving a base early is absolutely out if the defensive team just throws the ball to the base he left during the play. The formality of an appeal isn’t required in 99% of cases.
The appeals process is kind of weird, though it did make sense when they came up with it.
I had a similar situation while coaching a Little League playoff game last year.
Bottom of the 6th, 2 out, runners on 2nd and 3rd and the wrong batter comes to the plate and takes a 2-2 count. Suddenly, the offensive team yells at the batter that he is not supposed to be at bat and the umpire immediately calls the proper batter out. Game over, we win! Of course, afterwards, I realized that the umpire’s call was wrong… but it was too late.
From LL rulebook:
So the proper batter should have come to the plate with the 2-2 count and played on.
If I had noticed before the offensive team, in this situation, I would have intentionally walked to improper batter and immediately reported to the ump before the next pitch.
Exactly the same basis applies in cricket.
A batsman might be bowled middle stump, spoon a simple catch, get run out, or cause any of the other means of dismissal;
If the fielding team doesn’t appeal the umpire does nothing.
If the batsman remains at the crease and faces the next delivery, it’s play on.
A batsman isn’t technically out until they leave the field, even if there is an appeal and the umpire gives them out. Conversely if a batsman leaves the field (except at a break in play which is signaled by the umpire removing the bails) then they are out regardless of whether there has been an appeal or any decision by the umpire.
I think the OP has it right: Charlie is the proper batter in this situation. It seems weird, but the defensive team let it happen when they didn’t appeal after either Baker’s or Abel’s at-bat. This is a big screw-up by the defensive manager. He allowed two batters to reach base when either one could have been called out if he’d just paid attention.
Leaving a base early isn’t cheating. It’s a baserunning mistake that can get a runner called out. The fact that the defensive team has to take action to make this happen isn’t unique. Most outs happen because the defensive team took some type of action. What makes an appeal play different is that a runner can be called out after the fact for something he did (leave a base early, miss a base, bat out of order, etc.)